Problem: A circle has a sector with area $\dfrac{45}{2}\pi$ and central angle $225^\circ$. What is the area of the circle? ${36\pi}$ $\color{#9D38BD}{225^\circ}$ ${\dfrac{45}{2}\pi}$
Solution: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{225^\circ}{360^\circ} = \dfrac{45}{2}\pi \div A_c$ $\dfrac{5}{8} = \dfrac{45}{2}\pi \div A_c$ $A_c \times \dfrac{5}{8} = \dfrac{45}{2}\pi$ $A_c = \dfrac{45}{2}\pi \times \dfrac{8}{5}$ $A_c = 36\pi$